The average length of a trajectory in a certain billiard in a flat two-torus

TL;DR

This paper investigates the asymptotic behavior of the first exit time moments for a particle in a flat two-torus with a small removed disc, providing new estimates as the disc radius approaches zero.

Contribution

It offers novel asymptotic estimates for the moments of the first exit time in a billiard system on a flat torus with a small obstacle.

Findings

Asymptotic estimates for exit time moments as disc radius tends to zero

Quantitative understanding of particle trajectories in modified billiard systems

Insights into the dynamics of particles in perturbed flat tori

Abstract

We remove a small disc from the flat two-dimensional torus and consider a point-like particle that starts moving from the center of the disc with linear trajectory. We provide asymptotic estimates for the moments of the first exit time, with respect to the velocity, when the radius of the disc tends to zero.